Alternatives: Numerical solution for Karhunen Loeve expansion

Overview

The analytical solutions of the Fredholm equation only exist for some particular processes, that is, for particular covariance functions. For this reason the numerical solutions are usually essential. There are two main methods: the integral method and the expansion method.

Choosing the Alternatives

The eigenfunction is not explicitly defined when using the integral method.Therefore the expansion method is preferred where the eigenfunction is approximated using some orthonormal basis functions.

The Nature of the Alternatives

Integral method

This is a direct approximation for the integral, i.e. \(\int_{0}^{1}f(t)dt \approx \sum_{i=0}^{n+1}\omega_if(t_i).\)

The bigger the \(n\), the better the approximation.

Expansion method

This method is known as Galerkin method. It employs a set of basis functions. The commonly used basis functions are Fourier (trigonometric) or wavelets. Implementation is discussed in ProcFourierExpansionForKL and ProcHaarWaveletExpansionForKL. Note that although we have presented the methods separately, there are many common features which would comprise a generic method which could be used, in principle, for any approximation based on a particularly basis.